The [[Riemann Integral|Riemann integral]] is the common definition today of the integral, as first put on a rigorous mathematical foundation by Riemann. The Riemann integral is, in essence, the definite integral:

+

The [[Riemann Integral|Riemann integral]] is the common definition today of the integral in introductions to analysis, as first put on a rigorous mathematical foundation by Riemann. The Riemann integral is, in essence, the definite integral:

:<math>\int_a^b f(x)\,dx</math>

:<math>\int_a^b f(x)\,dx</math>

−

It is defined as the sum of infinitesimal segments of the area under the function on a graph, between ''f(a)'' and ''f(b)''.

+

It is defined as the sum of infinitesimal segments of the x-axis of the area under the function on a graph, between ''f(a)'' and ''f(b)''.

+

+

As it is more versatile, the [[Lebesgue]] Integral has superseded this definition.

==Additional concepts named after Riemann==

==Additional concepts named after Riemann==

Revision as of 18:15, 27 July 2018

Georg Friedrich Bernhard Riemann (1826-1866) is considered to be one of the greatest modern mathematician,[1] and is the author of the biggest unsolved problem: the Riemann hypothesis. "It may ... be truly said of Riemann that he touched nothing that he did not in some measure revolutionize" and that he was "[o]ne of the most original mathematicians of modern times."[2] More than a dozen diverse, prominent mathematical fields or structures are named after Riemann (see below). Yet his Christianity was most important to him of all:[3]

“

During his life, [Riemann] held closely to his Christian faith and considered it to be the most important aspect of his life. At the time of his death, he was reciting the Lord's Prayer with his wife and passed away before they finished saying the prayer.

”

Contents

Early life

Riemann was born into an impoverished family in Germany, a devoutly Christian son of a Lutheran pastor. He was homeschooled by his father, and was sent to attend a prominent school (Johanneum at Luneburg) at the age of 16. Riemann quickly became bored with the math class and asked the director for more advanced material. The director gave him most advanced math books (including Leonhard Euler's works and Adrien Marie Legendre's Theory of Numbers). Riemann mastered them in merely a few days.[4]

Riemannian geometry

Riemann created a new type of geometry, which was elliptical. Riemannian geometry then became the basis for advances in physics in the 20th century.

Most of the theorems of Riemannian geometry are different from their counterparts in Euclidian geometry. For example, the sum of the angles in a triangle in Riemannian geometry is greater than 180 degrees, which is the sum in Euclidian geometry. Also, there are no parallel lines in Riemannian geometry.

Riemann hypothesis

Riemann hypothesis remains the greatest unsolved problem in mathematics today. It was first proposed in Riemann's classic paper On the Number of Primes Less Than a Given Magnitude (1859).[5]

Riemann integral

The Riemann integral is the common definition today of the integral in introductions to analysis, as first put on a rigorous mathematical foundation by Riemann. The Riemann integral is, in essence, the definite integral:

It is defined as the sum of infinitesimal segments of the x-axis of the area under the function on a graph, between f(a) and f(b).

As it is more versatile, the Lebesgue Integral has superseded this definition.

Additional concepts named after Riemann

A long list of concepts central to mathematics today are named after Riemann. They include, among many others, the Riemann integral, the Riemann series theorem, the Riemann-Roch theorem, the Riemann-Lebesgue lemma, the Riemann surface, the Riemann sphere, the Riemann localization principle, and the Riemann mapping theorem.